Math 2.0

Differential geometry Definition

Osculating circle

Builds on Curvature of a curveRegular curveTangent vector of a curveCircle

Definition

Let γ ⁣:I→ℝ2\gamma \colon I \to \R^2 be a regular curve of class C2C^2 on an interval II, and let t0∈It_0 \in I. Let TT be its unit tangent vector, let NN be TT rotated by a right angle counterclockwise, and let kk be its signed curvature. Assume that k(t0)≠0k(t_0) \ne 0. The centre of curvature at t0t_0 is c=γ(t0)+N(t0)/k(t0)c = \gamma(t_0) + N(t_0) / k(t_0), and the radius of curvature is 1/∣k(t0)∣1 / \abs{k(t_0)}. The osculating circle of γ\gamma at t0t_0 is the circle with centre cc and radius 1/∣k(t0)∣1 / \abs{k(t_0)}.

Remarks

Example: for γ(t)=(t,t2)\gamma(t) = (t, t^2) at t0=0t_0 = 0, one has T=(1,0)T = (1, 0), N=(0,1)N = (0, 1) and k(0)=2k(0) = 2. So the osculating circle has centre (0,1/2)(0, 1/2) and radius 1/21/2. Non-example: for γ(t)=(t,t3)\gamma(t) = (t, t^3) at t0=0t_0 = 0, the curvature is 00, so there is no osculating circle there; the tangent line plays its role. The osculating circle passes through γ(t0)\gamma(t_0), has the same tangent there, and is the only circle with contact of order at least 22 with the curve. For a curve in ℝ3\R^3 with κ(t0)>0\kappa(t_0) > 0, the osculating circle is defined in the same way, with NN the principal normal. It lies in the osculating plane, spanned by TT and NN.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Curvature of a curve, concept, no evidence levelCurvature of a curveconceptEuclidean distance, concept, no evidence levelEuclidean distanceconceptEuclidean plane, object, no evidence levelEuclidean planeobjectEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptCircle, definition, no evidence levelCircledefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionRegular curve, definition, no evidence levelRegular curvedefinitionOsculating circle, definition, no evidence levelOsculating circledefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Curvature of a curve concept
  2. Euclidean distance concept
  3. Euclidean plane object
  4. Euclidean space object
  5. Function concept
  6. Circle definition
  7. Fréchet derivative definition
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Limit of a function of several variables concept
  11. Continuity in R^n definition
  12. Differentiable function of several variables definition
  13. Partial derivative definition
  14. Differentiable on an open set definition
  15. Higher partial derivatives definition
  16. Iterated derivative definition
  17. C^k function characterised by continuous derivatives statement
  18. Parametrised curve definition
  19. Tangent vector of a curve definition
  20. Regular curve definition
  21. Osculating circle definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Osculating_circle
About this page
TagZ2Z9BE946H
URIm2://Z2Z9BE946H
Kinddefinition
WritingW0 stub
MSC53A04
DefinesOsculating circle
UsesCurvature of a curve, Regular curve, Tangent vector of a curve, Circle
Changed2 November 2026

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